Finite Group Databasebeta

← C3×C5×#2122 #14275156

Subgroups — C3×C5×#2122 #14275156

Order 1920.

532 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#535C211yesmaximal, characteristic
2C3×C5×#413C211yesmaximal, characteristic, complemented
3C3×C5×#410C211yesmaximal, characteristic, complemented
4C3×C5×AΣL(1,𝔽16)C214yesmaximal, complemented
5C5×#2122C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×C5×#41C2211yescharacteristic
7C22×C3×C5×D4C411yescharacteristic, complemented
8C2×C3×C5×M16—21no—
9C3×C5×#127—21no—
10C3×C5×#124—21no—
11C3×C5×#121—21no—
12C3×C5×#121—21no—
13C3×C5×#120C412yescomplemented
14C3×C5×#120C2212yes—
15C3×C5×#117—21no—
16C3×C5×#117C411yescharacteristic, complemented
17C3×C5×#101—21no—
18C3×C5×#100C2212yes—
19C3×C5×#99—21no—
20C3×C5×#99C2212yes—
21C3×#2122C511yesmaximal, characteristic, direct product factor, Hall
22C5×#535C2×C311yescharacteristic
23C5×#413C2×C311yescharacteristic, complemented
24C5×#410C2×C311yescharacteristic, complemented
25C5×AΣL(1,𝔽16)C2×C314yescomplemented
26C24×C3×C5C2×C412yes—
27C3×C5×#41—41no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8D411yescharacteristic
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—41no—
35C2×C3×C5×D4—41no—
36C2×C3×C5×D4C2×C411yescharacteristic
37C2×C3×C5×D4D411yescharacteristic
38C2×C3×C5×D4C2311yescharacteristic
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×C4×C5D411yescharacteristic
41C22×C3×C4×C5D411yescharacteristic
42C3×C5×M16—22no—
43C3×C5×M16—41no—
44C2×C3×C5×C8—41no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—41no—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—22no—
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4C2×C412yes—
51C3×C42×C5—21no—
52C3×C42×C5—41no—
53C3×#535C2×C511yescharacteristic
54C3×#413C2×C511yescharacteristic, complemented
55C3×#410C2×C511yescharacteristic, complemented
56C3×AΣL(1,𝔽16)C2×C514yescomplemented
57C2×C5×#41C22×C311yescharacteristic
58C22×C5×D4C3×C411yescharacteristic, complemented
59C2×C5×M16—21no—
60C5×#127—21no—
61C5×#124—21no—
62C5×#121—21no—
63C5×#121—21no—
64C5×#120C3×C412yescomplemented
65C5×#120C22×C312yes—
66C5×#117—21no—
67C5×#117C3×C411yescharacteristic, complemented
68C5×#101—21no—
69C5×#100C22×C312yes—
70C5×#99—21no—
71C5×#99C22×C312yes—
72#2122C3×C511yescharacteristic, direct product factor, Sylow
73C23×C3×C5—22no—
74C23×C3×C5—42no—
75C23×C3×C5—82no—
76C23×C3×C5C22⋊C411yescharacteristic
77C23×C3×C5C22×C411yescharacteristic
78C23×C3×C5C2×D411yescharacteristic
79C3×C5×Q8—41no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4—41no—
83C3×C5×D4—42no—
84C3×C5×D4—41no—
85C3×C5×D4—41no—
86C3×C5×D4—82no—
87C3×C5×D4—82no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—42no—
96C2×C3×C4×C5C22⋊C411yescharacteristic
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C3×C5×C8—42no—
101C2×C3×#41C22×C511yescharacteristic
102C22×C3×D4C4×C511yescharacteristic, complemented
103C2×C3×M16—21no—
104C3×#127—21no—
105C3×#124—21no—
106C3×#121—21no—
107C3×#121—21no—
108C3×#120C4×C512yescomplemented
109C3×#120C22×C512yes—
110C3×#117—21no—
111C3×#117C4×C511yescharacteristic, complemented
112C3×#101—21no—
113C3×#100C22×C512yes—
114C3×#99—21no—
115C3×#99C22×C512yes—
116C24×C5C2×C3×C412yes—
117C5×#41—41no—
118C5×#41—41no—
119C2×C5×Q8C3×D411yescharacteristic
120C2×C5×D4—22no—
121C2×C5×D4—22no—
122C2×C5×D4—21no—
123C2×C5×D4—21no—
124C2×C5×D4—41no—
125C2×C5×D4—41no—
126C2×C5×D4C2×C3×C411yescharacteristic
127C2×C5×D4C3×D411yescharacteristic
128C2×C5×D4C23×C311yescharacteristic
129C22×C4×C5C2×C3×C411yescharacteristic
130C22×C4×C5C3×D411yescharacteristic
131C22×C4×C5C3×D411yescharacteristic
132C5×M16—22no—
133C5×M16—41no—
134C2×C5×C8—41no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4—41no—
137C5×C22⋊C4—22no—
138C5×C22⋊C4—22no—
139C5×C22⋊C4—22no—
140C5×C22⋊C4C2×C3×C412yes—
141C42×C5—21no—
142C42×C5—41no—
143#535C2×C3×C511yescharacteristic
144#413C2×C3×C511yescharacteristic, complemented
145#410C2×C3×C511yescharacteristic, complemented
146AΣL(1,𝔽16)C2×C3×C514yescomplemented
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5—21no—
150C22×C3×C5—41no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5—42no—
154C22×C3×C5—42no—
155C22×C3×C5—42no—
156C22×C3×C5—84no—
157C22×C3×C5C2×C22⋊C411yescharacteristic
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—42no—
162C3×C4×C5—82no—
163C3×C4×C5#9911yescharacteristic
164C3×C4×C5#9911yescharacteristic
165C24×C3C2×C4×C512yes—
166C3×#41—41no—
167C3×#41—41no—
168C2×C3×Q8C5×D411yescharacteristic
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—41no—
174C2×C3×D4—41no—
175C2×C3×D4C2×C4×C511yescharacteristic
176C2×C3×D4C5×D411yescharacteristic
177C2×C3×D4C23×C511yescharacteristic
178C22×C3×C4C2×C4×C511yescharacteristic
179C22×C3×C4C5×D411yescharacteristic
180C22×C3×C4C5×D411yescharacteristic
181C3×M16—22no—
182C3×M16—41no—
183C2×C3×C8—41no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—41no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4C2×C4×C512yes—
190C3×C42—21no—
191C3×C42—41no—
192C23×C5—22no—
193C23×C5—42no—
194C23×C5—82no—
195C23×C5C3×C22⋊C411yescharacteristic
196C23×C5C22×C3×C411yescharacteristic
197C23×C5C2×C3×D411yescharacteristic
198C5×Q8—41no—
199C5×D4—41no—
200C5×D4—41no—
201C5×D4—41no—
202C5×D4—42no—
203C5×D4—41no—
204C5×D4—41no—
205C5×D4—82no—
206C5×D4—82no—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5—21no—
211C2×C4×C5—21no—
212C2×C4×C5—21no—
213C2×C4×C5—22no—
214C2×C4×C5—42no—
215C2×C4×C5C3×C22⋊C411yescharacteristic
216C2×C4×C5C3×C22⋊C411yescharacteristic
217C2×C4×C5C3×C22⋊C411yescharacteristic
218C2×C4×C5C2×C3×D411yescharacteristic
219C5×C8—42no—
220C2×#41C22×C3×C511yescharacteristic
221C22×D4C3×C4×C511yescharacteristic, complemented
222C2×M16—21no—
223#127—21no—
224#124—21no—
225#121—21no—
226#121—21no—
227#120C3×C4×C512yescomplemented
228#120C22×C3×C512yes—
229#117—21no—
230#117C3×C4×C511yescharacteristic, complemented
231#101—21no—
232#100C22×C3×C512yes—
233#99—21no—
234#99C22×C3×C512yes—
235C2×C3×C5—21no—
236C2×C3×C5—41no—
237C2×C3×C5—41no—
238C2×C3×C5—41no—
239C2×C3×C5—82no—
240C2×C3×C5C2×#9911yescharacteristic
241C23×C3—22no—
242C23×C3—42no—
243C23×C3—82no—
244C23×C3C5×C22⋊C411yescharacteristic
245C23×C3C22×C4×C511yescharacteristic
246C23×C3C2×C5×D411yescharacteristic
247C3×Q8—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—42no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—82no—
255C3×D4—82no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—42no—
264C2×C3×C4C5×C22⋊C411yescharacteristic
265C2×C3×C4C5×C22⋊C411yescharacteristic
266C2×C3×C4C5×C22⋊C411yescharacteristic
267C2×C3×C4C2×C5×D411yescharacteristic
268C3×C8—42no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5—41no—
275C22×C5—42no—
276C22×C5—42no—
277C22×C5—42no—
278C22×C5—84no—
279C22×C5C2×C3×C22⋊C411yescharacteristic
280C4×C5—21no—
281C4×C5—21no—
282C4×C5—21no—
283C4×C5—42no—
284C4×C5—82no—
285C4×C5C3×#9911yescharacteristic
286C4×C5C3×#9911yescharacteristic
287C24C2×C3×C4×C512yes—
288#41—41no—
289#41—41no—
290C2×Q8C3×C5×D411yescharacteristic
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4C2×C3×C4×C511yescharacteristic
298C2×D4C3×C5×D411yescharacteristic
299C2×D4C23×C3×C511yescharacteristic
300C22×C4C2×C3×C4×C511yescharacteristic
301C22×C4C3×C5×D411yescharacteristic
302C22×C4C3×C5×D411yescharacteristic
303M16—22no—
304M16—41no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4C2×C3×C4×C512yes—
312C42—21no—
313C42—41no—
314C3×C5#212211yescharacteristic, direct product factor, Hall
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—84no—
325C22×C3C2×C5×C22⋊C411yescharacteristic
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—42no—
330C3×C4—82no—
331C3×C4C5×#9911yescharacteristic
332C3×C4C5×#9911yescharacteristic
333C2×C5—21no—
334C2×C5—41no—
335C2×C5—41no—
336C2×C5—41no—
337C2×C5—82no—
338C2×C5C2×C3×#9911yescharacteristic
339C23—22no—
340C23—42no—
341C23—82no—
342C23C3×C5×C22⋊C411yescharacteristic
343C23C22×C3×C4×C511yescharacteristic
344C23C2×C3×C5×D411yescharacteristic
345Q8—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—42no—
350D4—41no—
351D4—41no—
352D4—82no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—42no—
362C2×C4C3×C5×C22⋊C411yescharacteristic
363C2×C4C3×C5×C22⋊C411yescharacteristic
364C2×C4C3×C5×C22⋊C411yescharacteristic
365C2×C4C2×C3×C5×D411yescharacteristic
366C8—42no—
367C2×C3—21no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—82no—
372C2×C3C2×C5×#9911yescharacteristic
373C5C3×#212211yescharacteristic, direct product factor, Sylow
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—84no—
384C22C2×C3×C5×C22⋊C411yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—82no—
390C4C3×C5×#9911yescharacteristic
391C4C3×C5×#9911yescharacteristic
392C3C5×#212211yescharacteristic, direct product factor, Sylow
393C2—21no—
394C2—41no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
center, socleC2×C3×C5C2×#99
commutatorC23C22×C3×C4×C5
Frattini, cosocleC2×D4C23×C3×C5
prime radicalC3×C5×#535C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×C5×#413C2semidirect
3. C3×C5×#410C2semidirect
4. C3×C5×AΣL(1,𝔽16)C2semidirect
5. C5×#2122C3direct (N × H)
7. C22×C3×C5×D4C4semidirect
13. C3×C5×#120C4semidirect
16. C3×C5×#117C4semidirect
21. C3×#2122C5direct (N × H)
23. C5×#413C2×C3semidirect
24. C5×#410C2×C3semidirect
25. C5×AΣL(1,𝔽16)C2×C3semidirect
54. C3×#413C2×C5semidirect
55. C3×#410C2×C5semidirect
56. C3×AΣL(1,𝔽16)C2×C5semidirect
58. C22×C5×D4C3×C4semidirect
64. C5×#120C3×C4semidirect
67. C5×#117C3×C4semidirect
72. #2122C3×C5direct (N × H)
102. C22×C3×D4C4×C5semidirect
108. C3×#120C4×C5semidirect
111. C3×#117C4×C5semidirect
144. #413C2×C3×C5semidirect
145. #410C2×C3×C5semidirect
146. AΣL(1,𝔽16)C2×C3×C5semidirect
221. C22×D4C3×C4×C5semidirect
227. #120C3×C4×C5semidirect
230. #117C3×C4×C5semidirect
314. C3×C5#2122direct (N × H)
373. C5C3×#2122direct (N × H)
392. C3C5×#2122direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C5×#5352
C3×C5×#4132
C3×C5×#4102
C3×C5×AΣL(1,𝔽16)2
C5×#21223
C3×#21225